Theorem prnz 4734

Description: A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)

Hypothesis

Ref	Expression
prnz.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
prnz	⊢ {𝐴, 𝐵} ≠ ∅

Proof of Theorem prnz

Step	Hyp	Ref	Expression
1		prnz.1	. . 3 ⊢ 𝐴 ∈ V
2	1	prid1 4719	. 2 ⊢ 𝐴 ∈ {𝐴, 𝐵}
3	2	ne0ii 4289	1 ⊢ {𝐴, 𝐵} ≠ ∅

This proof depends on syntax axioms:   ∈ wcel 2144   ≠ wne 2955  Vcvv 3450  ∅c0 4278  {cpr 4586

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-v 3452  df-dif 4047  df-un 4151  df-nul 4279  df-sn 4585  df-pr 4587

This theorem is used by:  opnz  5441  propssopi  5479  fiint  9321  wilthlem2  27401  upgrbi  29676  wlkvtxiedg  30210  shincli  31969  chincli  32067  constrextdg2lem  34426  spr0nelg  48651  sprvalpwn0  48658